Math 720: Measure Theory and Integration

This is a first graduate course on Measure Theory, and will at least include the following.

• Outer measures, measures, $\sigma$-algebras, Carathéodory's extension theorem.
• Borel measures, Lebesgue measures.
• Measurable functions, Lebesgue integral (Monotone Convergence Theorem, Fatou's Lemma, Dominated Convergence Theorem).
• Modes of Convergence (Egoroff's Theorem, Lusin's Theorem)
• Product Measures (Fubini-Tonelli Theorems), $n$-dimensional Lebesgue integral
• Signed Measures (Hahn Decomposition, Jordan Decomposition, Radon-Nikodym Theorem, change of variables)
• Differentiation (Lebesgue Differentiation Theorem)
• $L^p$ Spaces, Hölder's inequality, Minkowskii's inequality, completeness, uniform integrability, Vitali's convergence theorem.

This will be followed by some special topics (e.g. Fourier Analysis).

Course Outline

• The course will start by constructing the Lebesgue measure on $\mathbb{R}^n$, roughly following Bartle, chapters 11--16.
• After this, we will develop integration on abstract measure spaces roughly roughly following Cohn, chapters 1--6.
• If time permits, I will continue with some Fourier Analysis roughly following Folland chapter 8.
• I recommend buying Cohn. If you can, I would also recommend buying Folland, as it additionally has a little bit of topology, functional analysis, Fourier analysis and probability (the book however has many typos).

Alternate references

• Another excellent (but harder) reference is Rudin, which I strongly recommend.
• Alternately, two slightly easier books are Jones or Royden.
• If you prefer learning from lecture notes, here are some by Lenya Ryzhik and Terry Tao. (The last one is available as a PDF, and also as a regular published book.) Alternately, contact Giovanni Leoni for last years measure theory lecture notes.
• An excellent treatment of Fourier Series can be found in Chapter 1 of Wilhelm Schlag's notes. (This has many advanced Harmonic Analysis topics, which I recommend reading later.)